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Here are three e-mails I received from three different people over the last three months. Spot the common theme.

November:

My co-teacher and I were puzzling over what kind of problem would create an intellectual need for systems. Do you have anything you could send, by chance?

December:

We are planning to launch a unit on systems of equations in early January (after December break) and wanted to try out your approach to create an intellectual “need.”

January:

Showing two straight lines on a piece of graph paper and finding points of intersection has very little significance to most people. I’m looking for a real-world problem that has an answer that is not self-evident, but which requires a little thinking and finding the intersections and is infinitely more productive and satisfying and will stay with them for the rest of their lives. That is what I am looking for.

I receive these questions on Twitter also. I find them almost impossible to answer because I don’t know what your class worships.

Here’s what I’m talking about:

Class #1

You start class by asking your students to write down two numbers that add to ten. They do. Most likely a bunch of positive integers result.

Then you ask them to write down two numbers that subtract and get ten. They do.

Then you ask them to write down two numbers that do both at the exact. same. time. “Is that even possible?” you ask.

Many of them think that’s totally impossible. You can’t take the same two numbers and get the same output with two operations that are natural enemies of each other. They’d maybe never phrase it that way but the whole setup seems totally screwy and counterintuitive.

Then someone finds the pair and it seems obvious in hindsight to most students. We’ve been puzzled and now unpuzzled. Then you ask, “Is that the only pair that works?” knowing full well it is, and the class is puzzled again.

You define systems of equations as “finding numbers that make statements true” and you spend the next week on statements that have only one solution, that have infinite solutions, and the disagreeable sort that don’t have any solutions.

Students learn to identify the kind of scenario they’re looking at and how to find its solutions quickly (if any exist) using strong new tools you offer them over the unit.

Class #2

The same lesson plays out but this time, after we’ve determined the pair of numbers that solve the system, a student pipes up and asks, “When will we ever use this in the real world.”

Worshipping the Real World

David Foster Wallace wrote about worship – the secular kind, the kind that applies to everybody, not just the devout, the kind that applies especially to us teachers in here:

If you worship money and things — if they are where you tap real meaning in life — then you will never have enough. Never feel you have enough.

If your students worship grades, they won’t complete assignments without knowing how many points it’s worth. If they worship stickers and candy, they won’t work without the promise of those prizes.

If you say a prayer to the “real world” every time you sit down to plan your math lessons, you and your students will never have enough real world, never feel you have enough connection to jobs and solar panels and trains leaving Chicago and things made of stuff.

If you instead say a prayer to the electric sensation of being puzzled and the catharsis that comes from being unpuzzled, you will never get enough of being puzzled and unpuzzled.

The first prayer limits me. The first prayer means my students will only be interested in something like The Slow Forty – a real world application of systems. The second prayer means my students will be interested in The Slow Forty (because it’s puzzling) but also the puzzling moments that arise when we throw numbers, symbols, and shapes against each other in interesting ways.

The second prayer expands me. Interested people grow more interested. Silvia writes, “Interest is self-propelling. It motivates people to learn thereby giving them the knowledge needed to be interested” (2008, p. 59). Once you give your students the experience of becoming puzzled and unpuzzled by numbers, shapes, and variables, they’re more likely to be puzzled by numbers, shapes, and variables later. That’s fortunate! Because some territories in mathematics are populated exclusively by numbers, shapes, and variables, in which cases your first prayer will be in vain.

That’s why I can’t tell you what to teach on Monday. Your classroom culture will beat any curriculum I can recommend. I need to know what you and your students worship first.

26 Responses to “[Fake World] Culture Beats Curriculum”

I can’t tell you how uncomfortable my students are with being puzzled. So much so that their first instinct is to ask The Internet (or Google, or Siri, or what have you). As you’ve famously said in the past, they have NO patience for irresolution. That’s a big problem.

As a science teacher I can agree that real world does not equal automatic interest. Science is only about the real world, and yet science class is not instantly engaging because of it. I think the general problem science and math education have is that we put the answer before students have the question. Students aren’t supposed to wonder, “Why do the days get shorter in winter?” we are supposed to tell them why, whether they wonder it or not, because we are on that chapter.

I’ve been reading all the fake world posts and wondering where it was going to end. It always seemed obvious to me that “real world” wasn’t the panacea for making math engaging. Sometimes it can be and sometimes it won’t be.

In this post, you nailed it! Well written and I couldn’t agree more that being puzzled and unpuzzled is where it’s at.

I think the what we should be trying to figure out is how to get kids to embrace and enjoy the puzzled/unpuzzled mindset. Making puzzled/unpuzzled part of our classroom culture can be a challenge.

I’ve never read that quote by David Foster Wallace before, and I am glad you shared. Ultimately classroom culture is the most important factor. It impacts everything from instruction to learning to discussion to assessment. Everything.

few comments I wanted to make:

1. Developing a learning culture is important and difficult

Each student comes with their own prior experiences and values. At times it comes with parental pressures and impressions as well. But ultimately a classroom culture is the combined/integrated product of every student, parents, school, and teacher.

“If you say a prayer to the “real world” every time you sit down to plan your math lessons, you and your students will never have enough real world”

They are questions worth thinking about though. How exactly do we promote (since we can influence, but not create) these cultures within our classrooms?

As Matt E pointed out: students “have no patience for irresolution.” This would definitely influence how we choose to tackle the problem of changing existing classroom culture.

2. What we claim to worship, and what we actually worship

This may be self-explanatory. A while ago during a workshop we talked about whether our classroom environments actually reflect what we value as educators. Are your seats in rows? What does that say about what you value? Is there a board in the back displaying student products? What does that say about what you value? Are there lots of manipulatives in the room? What does that say? How much time do you spend talking versus how much time do students spend talking?… the list goes on. Intentionally or unintentionally – what we say we value may not completely reflect in how we have engineered our classroom environments. This applies to the notion of worship as well. After that workshop, I thought hard about how the environment can be used to work towards or against our values.

3. How much should we change or influence the culture?

This point actually slightly contradicts my first comment. How much should we change or influence the classroom culture?

A class loves mathematics that are applicable to the “real world.” Do we try our best to also convince them that the “fake world” is equally interesting? Actually, the quick answer is probably “yes.” But I guess a better question is how do we best approach this for students that jump away in horror every time anything “non-realistic” pops out at them? Alternatively, students that love the “fake world” and believes the “real world” is boring – we need to also expose them to the complexities of reality. How do we best do this? How often do we do this?

I do believe we should explore these types of questions on our own and arrive at our own answers for each different classes.

Some thoughts: Perhaps the mindset that every concept, every lesson, every skill, needs a real-world application to be exciting and motivating is the pendulum swinging in the opposite direction from, “do pgs 34 and 35 in the workbook.” But release from that mindset is liberating, actually. The puzzling/unpuzzling paradigm can applied to many learning experiences, and real-world is only one of them. From an elementary perspective, our curriculum is full, at every level, with all kinds of math games. Many are written into the curriculum. One alternative to real-world may exist in doing more to explore and mine the games we play. (I’m currently working on a post about what a colleague and I did with “Factor Captor”.) Kids love playing games, and taking them apart and seeing how they work, thinking about how to redesign them and reuse them, can yield some exciting results.
For the review assignment, I see a nice tension between the novel and the familiar.
I love the Wallace quote. I came across that many years ago and have never forgotten it. It comes from a commencement speech he gave at Kenyon College in 2005, and I would encourage everyone to take the time and read the whole thing. Yes, what we worship in our classrooms is critical. I think this is related to the paper you referenced a few months ago, “Intellectual Need and Problem-Free Activity in the Mathematics Classroom”. I now cringe whenever I hear a teacher say, “You need to know this for the test.” As an aside, I might add that the overwrought worship of “data” by the educational establishment has, and continues to have, very corrosive effects, and not only in math classrooms.

The students who are the least comfortable with this process are those who have always had an A and expect to “get it” quicky. Their parentsa re very uncomfortable with this as well and raise the biggest stink when this happens in their kids’ classrooms.

I have been enjoying the many discussions through these posts and wanted to offer a solution to the systems and intersection with the real world question.

I begin with the questions “who owns a car” and “who wants to own a car.” These are followed up with” are you buying new or used? what was the price of the used car when it was new? Why did the value decrease? does it decrease over time and if so, by how much?” These are the stimulation of interest questions. Fortunately, car ownership is high interest. We then move to the real life example applying a few students’ personal experiences in car ownership.

The lesson moves to determining the purchase price, the estimated life of a car, with the theoretical zero value at the end of a specified number of years.

Graph the points as purchase price, y intercept and zero value number of years as x intercept.

Now students can see the transformation of Cartesian coordinates to category axis and real life numbers transform back to (x,y) point values.

Students apply the billions (ha,ha) of slope calculations they have practiced to achieve the loss of value per year, slope.

Now take them on the journey of purchasing a vehicle with the use of a down payment and a monthly loan. They will now attempt to create the expense function and graph to with the depreciation function.

Students will trip and fall on the mixed units for the time axis. The depreciation is in years ad the expense function is in months. I let this slide and actually hope it is the case. This creates a create investigation after the students have solved the system of equations, i.e. breakeven of payout versus value. When students a re asked to prove the system solution by substituting back in to the equations, the looks on their faces tells the story and the detective work begins. hey will figure out that the expense equation must be converted to yearly loan payback or the depreciation equation must be converted to monthly rather than yearly units for slope.

You can use this model of systems to also graph exponential depreciation against expense functions,

AS well, you can use a table of depreciated care values form a Kelly Blue Book search to have students construct the basis of the lesson. Have students choose a car they would like to purchase and investigate the historical values, plot the values, run a linear regression, and for advancing skills, quadratic regression to determine best fit functions.

In the end, the discussion of why people trade in one or two year old vehicles is a great lesson to go back to the graph and explore exactly what breakeven means as the two functions converge.

If you wan tot have a great anchor problem for an assessment, blend loan payments formulas for student driven monthly payments. To challenge the “need to be challenged” DI group, you may wan to supply a break even point and challenge them to solve for the monthly payment amount and then remove a variety of variable values from the payment formula, i.e. interest rate, or length of loan payback. You may also add complexity with cost of ownership data.

I have had very good success and high levels of engagement with this investigation. In the end, students tend to value the understanding of how to make decisions will data and arithmetic modeling (Quantitative Reasoning), as well as understanding why their parent of a friend’s parent trades in a one or two year old car for a new one.

This is pretty closely related to why I’m scared at the thought of things like rolling out SBG as a top-down mandate. The best assessment strategy ever can still turn into mush if your beliefs about teaching aren’t aligned with it.

Kevin Kelly http://www.edge.org/conversation/the-technium
says something about the future of technology that can just as well be applied to Mathematics: “…real creativity comes when you’re wasting time and when you’re fooling around without a goal. That’s often where real exploration and learning and new things come from…Think of [these obsessions] as kind of a compulsion that is trying to explore something, and right now there’s a compulsion about what can you do with 140 characters, and we’ll work through that and we’ll say, ”
That’s why the fake compulsion of “real world” doesn’t work. It’s stifling.

You have successfully avoided answering a question by providing a false answer to a false question.

Classrooms do not always “worship” something. Even if they do, can’t it be both or if it’s the second, can’t they desire the first? Some people value purpose and you’re taking the purpose away from them and their desire to learn.

If you try and create an environment where your students needs are ignored because you’re pointing them in the direction you want them to go, you will soon find that they go in their own direction which does not follow your curriculum. Not everyone has a passion for math.

@matthew Kennedy: “Some people value purpose and you’re taking the purpose away from them and their desire to learn.”

Game designers/developers (my former profession) would argue that a balance of challenge and skill — driven by even the simplest “puzzle” — IS a purpose, and one of the most compelling. Besides enabling the psychology of “optimal experience”, known as the Flow State, a balance of progressively increasing challenge and skill creates a reinforcing loop that drives the kind of motivation/engagement we all see in kids and adults playing games.

A look through the *themes* of most games — even the most addictive — we notice a dramatic absence of “real world” purpose. But brains are tuned to find even the most arbitrary puzzle challenges irresistible under the right circumstances.

We know from what kids do voluntarily *outside* of school that students do NOT need a “passion for math” OR a “real world purpose”. They just need a brain. But the key word is “voluntary”. The psychology of motivation (Self-Determination Theory, in particular — the leading theory of intrinsic motivation) finds a strong connection between motivation and autonomy. Feeling autonomous need not mean we have total independence, but that what we are doing is what we *wanted* to do, even if someone else asked us to do it.

Finding puzzling challenges that tickle the brain into responding, regardless of the theme/”purpose”, is not trivial, but certainly do-able. Finding a way to create a context of autonomy — the context where the intrinsic motivation of puzzle/challenges thrives — is (I think) an even tougher problem for teachers and parents.

Just one of the reasons I’ve always found Dan’s work so useful is that it combines what makes games compelling (balance of challenge and skill — “Flow”) with what sucks us into a movie or novel (including those with no obvious application to our “real life”). It all begins with a question for which your brain now can’t let it go. Of course students today do not by default view a classroom as a film or game, but that doesn’t mean the same principles don’t/can’t work.

I am totally jazzed by probing the underlying thoughts and beliefs that underpin one’s approach to teaching math. I’ve come at it intuitively with Horse Lover’s Math. Dan is putting into words what I’ve only sensed.

Kids are naturally curious about the world. At a certain point and in certain situations, that gets shut down. The “the electric sensation of being puzzled and the catharsis that comes from being unpuzzled” is about being awake, aware and totally present in the moment – alive.

I am eager to bring a new, clearer understanding and a deeper, more conscious perspective to HLM and the kids I work with in the classroom.

[…] months about what “real world” education means came to a spectacular head the other day in a post that veered through curiousity research, solvable and unsolvable equations, and David Fost…. Talking about DFW’s take on secular worship — that the worship of the scarce […]

I had students ask “when will we use this in the real world?” immediately following a unit where we did a project about budgets and bill paying where they were assigned given paycheck amounts and finding housing and paying bills based on their paycheck / life scenario.

Many kids will ask this question just to ask it regardless of how relevant the information is.

I love this post because it gets to the beating heart of everything— the distinction, as Jim P. puts it, between what we (and our students) SAY we value and what we (and our students) ACTUALLY value. You have to open up to the reality of what is ACTUALLY HAPPENING RIGHT NOW in order to value — and even to notice — whether or not the intellectual need in the room is genuine.

The only thing I would add is that for me, the binary distinction of puzzled/unpuzzled is too limited. It’s more goal-directed than curiosity can be. There is a kind of ecstasy to simply noticing what is going on (and its attendant “becoming worshipfully curiosity”) that is intrinsically motivating in itself. I notice that I don’t always CARE about the goal of becoming unpuzzled if the experience of puzzlement truly captivates me.

Students can be curious about all kinds of things beyond the physics of the “real world.” Thanks for this reflection.

[…] discussion ranks very highly. That could be peer instruction for physics. It could be one of Dan Meyer’s puzzlers. It could be your Socratic dialogue on the strengths and weaknesses of democratic […]

This week I put a “puzzle” up for students to do during a break for which I didn’t have the answer – and I told them this. Some students, who usually ‘get it quickly’ got frustrated when they couldn’t work out the answer straight away, some stayed and some walked away. One student came the next day and said “Miss I spent 2 hours on that problem last night and I still haven’t got the answer” :) , another student spent the 2nd lesson of the double working it out on the whiteboard and still didn’t have an answer but you could see he thoroughly enjoyed the challenge of finding links in the maths that he knew to try and find the answer.

My statement during this was is it’s not the “answer” that is the most important but the journey of finding out what works and what doesn’t. Everyone can say ‘Oh I should have done this and that’ with hindsight.